Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)

  1. For each unital -algebras, , .
  2. If and are unital -algebras, and if and are -homomorphisms, then
  3. .
  4. For every pair of -algebras and , .
    Parts (1) and (2) say that is a functor from the category of unital -algebras to the category of Abelian groups. Parts (3) and (4) say that maps zero objects to zero objects, and zero morphisms to zero morphisms. WE INCLUDE THE ZERO C-ALGEBRA IN THE RANKS OF UNITAL C-ALGEBRAS!
    Proof:
    Use the definition of the functor
  5. Take a map at the level of the algebras, , a * -homomorphism.
  6. Induce a map on the matrix algebras of each one, .
  7. Recall that it sends projections to projections and so it induces another map .
  8. Lastly define a map from which after being put inside of the grothendieck map has the same properties as the universal property of as seen in Proposition 3.1.8 (Universal property of K0).
  9. This forces it to factor through a unique map we denote as
    So now we check that .
    As desired.
Now lets think about why